What should I do about my mole? (Photo)

It sounds like you have a seborrheic keratosis.   It's hard to rip an actual mole.  As to the question you asked, what to do, that is simple.   See a doctor.

Trauma or irritation can certainly cause normal moles to bleed.  We see this all the time.  However, bleeding should always raise some concern and prompt further investigation.  As skin cancers grow, they are often more fragile than normal skin.  These skin cancers may bleed spontaneously (without a reason) or bleed more easily from relatively light trauma.  For example, some patients notice a skin cancer on their face because it always seems to bleed when they dry off with a towel after taking a shower, or easily bleeds when shaving their face.  Of course, you can also cut a normal mole while shaving!  
If you have a bleeding mole or spot, the smartest option is to have it checked by a skin specialist, particularly a dermatologist or plastic surgeon.  Hopefully they can offer you some reassurance.  Even if the mole is benign, sometimes it is wise to go ahead and have it removed...it may get snagged and ripped again!  Removal is usually easily performed with a small amount of local anesthesia.  The spot can be simply shaven off flat with the skin surface, and a band-aid applied for a few days with a dab of antibiotic ointment.  On some occasions, a deeper cut and stitches are preferred.  Usually we recommend sending the spot to the pathology laboratory to confirm the diagnosis, and hopefully to give you peace of mind.